La legge gravitazionale dell'Universo del quadrato nei principi di Newton

We revisit the question concerning the accuracy of the inverse-square gravitational law, as raised in section IX, book I of Newton’s Principia, and in some sections of book III. We argue that the problem has been thoroughly discussed and basically solved by Newton himself. We point out in particular the relations with the well-known Bertrand’s theorem. Bertrand approached the same problem in moreabstract terms. We remark, however, that much of Bertrand’s proof is essentially contained in Principia.